Question : A hemisphere and a cone have equal bases. If their heights are also equal, then the ratio of their curved surfaces will be:
Option 1: $1 : 2$
Option 2: $2 : 1$
Option 3: $1:\sqrt{2}$
Option 4: $\sqrt{2}:1$
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Correct Answer: $\sqrt{2}:1$
Solution :
Since the bases of the hemisphere and cone are equal, their radii are also equal.
⇒ radius of hemisphere = radius of cone = $r$
Height of hemisphere = radius of hemisphere = $r$
Also, height of cone = height of hemisphere = $r$
The slant height of the cone, $l$ = $\sqrt{r^{2}+r^{2}} = \sqrt{2}r$
Curved surface area of a hemisphere $= 2\pi r^{2}$
The curved surface area of a cone $= \pi rl$
Require ratio $= 2\pi r^{2}:\pi rl$
$= 2\pi r^{2}:\pi r\sqrt{2}r$
$= \sqrt{2}:1$
Hence, the correct answer is $\sqrt{2}:1$.
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